Why Are Significant Figures and Exact Numbers Important in Physics?
What are Significant Figures?
Significant figures of a number are difits that takes useful contribution to the resolution of measurement Dimensions are the technique of interconnecting with physical quantities in physics. A numeric value is yielded by each measurement.
As a number given below, each digit is crucial for the measurement process. Consequently, a few digits are more significant than others. Now, let’s study some of these significant digits.
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What are Significant Digits?
The objective of significant figures for numbers is to take the meaning which contributes to its measurement resolution.
The number 14.3 is included with three significant digits. All the time significant digits are known as the non-zero digits.
6.14134 possess 6 significant digits. Here, all the numbers offer useful information. Also, 59 have two significant digits, and 78.3 have three significant digits.
1000 has only one significant digit as the one is remarkable; we don’t recognize anything certainly about the units, tens, and hundreds of places, but the place holders are the zeroes in that number.
It is also the same with the number having a decimal given as 0.00028, which contains 2 significant digits i.e., only the 2 and 8 tell us something. The total availability of zeros is only the placeholders, and help to aid the information about approximate size.
Two thousand five (2005) has 4 significant digits i.e. the two and five are significant and we need to sum the zeroes as they’re between 2 the significant digits.
What are the Rules for Significant Figures?
The number of significant figures can be determined in a number by using these 3 rules:
It the zeros are between two significant digits will also significant
In the decimal, a last zero or behind zeros share is significant only
Non zero digits are always significant
1. Rules for Addition and Subtraction Use are as Follows:
Add or subtract in the standard way
In the decimal, portion calculates the number of significant figures only of each number in the problem
To the right of the decimal, the final answer may have no extra significant figures than the MINIMUM number of significant figures in any number in the problem.
2. Rules for Multiplication and Division are as Follows:
The MINIMUM amount of significant figures in any number of the problem governs the number of significant figures in the answer.
It implies that you have to distinguish significant figures to use this rule.
Significant Figures Errors in Measurements
1. Systematic Errors
This type of error arises from a fault in the measurement system, which frequently happens each time while a measurement is completed.
You must elude the same thing by doing it wrong each time during the measurement, i.e., your calculation will fluctuate systematically (which means each time it is in the same direction) from the accurate outcome.
2. Absolute and Relative Errors
If there is the uncertainty in the number and having similar units as the number itself, then it is known as the absolute error in a measured quantity.
For illustration, if you distinguish a length is 0.658 m ± 0.003 m; then the absolute error is 0.003 m.
The relative error is also known as the fractional error. It is found by dividing the absolute error in the number by the number itself. This kind of error is ordinarily more significant than the absolute error.
3. Random Errors
This kind of error ascends from the variations that are most simply perceived by creating multiple trials of a specified measurement.
For illustration, if you were to measure a pendulum’s period numerous times with the help of a stopwatch, you would discover that your measurements are not always identical.
What is the Correct Number of Significant Figures?
The correct number for the significant figures can be explained as significant numbers. It lies between 0 to 9 and is utilized as the coefficient of an expression clarifying the exactness of expression.
Some of the following numbers are given in the table for illustrations with decimal notation, scientific notation as well as significant figures:
What is Significant Figures Example?
Significant figures examples can be assimilated into a gradient of rules for significant figures;
Zeroes that lie between non-zero digits are significant: 2003 kg has 4 figures.
Entire non-zero digits are significant: 5.231g has four significant figures.
Following zeroes, which are also to the right of a decimal point in a number are significant: 0.0350 mL has three significant figures.
Foremost zeros to the left of the first non-zero digits are not significant.
When a number finishes in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant: 260 kilometers, maybe 2 or 3 significant figures.
FAQs on Error, Significant Figures, and Exact Numbers in Physics
1. What are significant figures in Physics, and why are they important for measurements?
In Physics, significant figures are the digits in a number that are reliable and necessary to indicate the quantity of something. They include all certain digits plus one final uncertain digit. Their importance lies in representing the precision of a measurement. For example, measuring a length as 5.2 cm (two significant figures) is less precise than measuring it as 5.23 cm (three significant figures), as the latter implies a more sensitive measuring instrument.
2. What are the key rules for determining the number of significant figures in a measurement?
To count the number of significant figures in a number, you can follow these primary rules as per the CBSE 2025-26 syllabus:
- All non-zero digits are always significant. (e.g., 12.345 has five significant figures).
- Any zeros between two non-zero digits are significant. (e.g., 70.05 has four significant figures).
- Leading zeros (zeros before non-zero digits) are not significant. (e.g., 0.0045 has two significant figures).
- Trailing zeros in a number with a decimal point are significant. (e.g., 2.500 has four significant figures).
- Trailing zeros in a whole number without a decimal point are ambiguous. To avoid this, numbers are written in scientific notation. For example, 5800 m can be written as 5.8 x 10³ (2 sig figs) or 5.800 x 10³ (4 sig figs).
3. How do you apply significant figure rules in calculations involving addition or subtraction?
When adding or subtracting measured quantities, the result should be rounded to the same number of decimal places as the measurement with the fewest decimal places. For example, if you add 12.11 cm, 18.0 cm, and 1.012 cm, the sum is 31.122 cm. However, since 18.0 cm has only one decimal place, the final answer must be rounded to 31.1 cm.
4. What is the rule for significant figures when multiplying or dividing two numbers?
For multiplication or division, the final result should have the same number of significant figures as the original number with the least number of significant figures. For instance, if you multiply a length of 4.52 cm (3 significant figures) by a width of 2.1 cm (2 significant figures), the calculator result is 9.492 cm². Since the least precise measurement has only two significant figures, the answer must be rounded to 9.5 cm².
5. How do exact numbers affect calculations involving significant figures?
Exact numbers are numbers that are not obtained by measurement and are known with complete certainty. Examples include numbers from definitions (like 1 minute = 60 seconds) or from counting discrete objects (like 5 books). Because they have no uncertainty, exact numbers are considered to have an infinite number of significant figures. Therefore, they do not limit the number of significant figures in a calculation. You only consider the measured values to determine the final precision.
6. What is the difference between accuracy and precision, and how do significant figures relate to them?
Accuracy and precision are two fundamental concepts in measurement:
- Accuracy refers to how close a measured value is to the true or accepted value.
- Precision refers to how close multiple measurements of the same quantity are to each other, regardless of their accuracy.
Significant figures are directly related to precision. A higher number of significant figures indicates a more precise measurement because it implies the use of a more sensitive instrument that can resolve smaller differences.
7. Why does rounding a final answer to the correct number of significant figures not reduce its actual scientific value?
Rounding a final answer is crucial because it correctly communicates the uncertainty of the result. Reporting a calculated answer with more digits than justified by the original measurements is misleading, as it implies a level of precision that was never achieved. A scientist or engineer understands that a chain is only as strong as its weakest link; similarly, a calculation is only as precise as its least precise measurement. Rounding ensures the final answer honestly reflects this limitation.
